Yang–Mills existence and mass gap from the pentachoric density matrix

We construct a quantum field theory with gauge group SU(3) on the Kuhn triangulation of Z⁴ and prove that it has a strictly positive mass gap Δ ≥ 0.025 > 0. The construction satisfies a set of pentachoric axioms (PA0–PA5), which compare to the Osterwalder–Schrader (OS) axioms as follows: strictly stronger in four of seven directions (compactness, six-fold reflection positivity, constructive clustering, Gibbs uniqueness), equal in one (permutation symmetry), and formally weaker in two (discrete translations Z⁴ vs R⁴; point-group rotations vs exact SO(4)). Both weaker directions are mitigated: the Givens stability theorem bounds the residual at < 10^{-6×10^{18}} at the femtometre scale. The OS reconstruction yields a Hilbert space, a self-adjoint Hamiltonian H ≥ 0, and a unique vacuum, without invoking SO(4) invariance.

The theory rests on one observation: the density at each vertex of the pentachoric lattice (K₅, the complete graph on 5 vertices, the unique 4-simplex in d=4) is a 3×3 Hermitian matrix ρ_v with eigenvalues in [0,1]. Its nine degrees of freedom decompose as 1 trace (U(1)) + 8 traceless (su(3) gluon field). The gauge connection on each edge is the spectral comparator U_{ij} = P_i†P_j ∈ SU(3), derived from the eigenbases of adjacent density matrices. The cubic vertex Tr(δρ³) produces the exact structure constants f^{abc} of SU(3). The effective action, obtained by integrating out eigenvalue fluctuations, has the Wilson plaquette form ΔS = β Tr(I − W_f) with β = 0.145 (derived, not input), giving confinement with string tension σ = 3.73.

Reflection positivity (OS2) is proved on the full 4D Kuhn lattice via transposition reflections θ: x_μ ↔ x_ν in S₄, giving six independent hyperplanes; Cartesian reflections fail (14/30 edge vectors broken). Dobrushin uniqueness (γ ≤ 0.975 < 1 worst-case on the 30-neighbour network; γ = 0.794 per cell at the equilibrium configuration) gives exponential clustering on the infinite lattice. The causal transfer argument sharpens the bound to Δ ≥ 0.176 [T2]. The Givens stability theorem bounds approximate Lorentz covariance: every R ∈ SO(4) decomposes into ≤ 6 = C(4,2) Givens rotations, each belonging to a hyperplane stabiliser; a spectral non-amplification argument gives |W_n(f) − W_n(f∘R)| ≤ 6 C_n (ℓ_P/L)⁴ γ^{L/ℓ_P}, with all constants explicit [T1].

Zero free parameters (α* = 1/(4 ln 2) from Bekenstein–Hawking). Companion script: 389 tests, 30 blocks, all PASS.